Lie powers of the natural module for GL(2,K)

Abstract

In recent work of R. M. Bryant and the second author a (partial) modular analogue of Klyachko's 1974 result on Lie powers of the natural $\mathrm{GL}(\mathrm{n},\mathrm{K})$ was presented. There is was shown that nearly all of the indecomposable summands of the $r$th tensor power also occur up to isomorphism as summands of the $r$th Lie power provided that $r\ne p^m$ and $r\ne 2p^m$, where $p$ is the characteristic of $K$. In the current paper we restrict attention to $\mathrm{GL}(2,K)$ and consider the missing cases where $r=p^m$ and $r=2p^m$. In particular, we prove that the indecomposable summand of the $r$th tensor power of the natural module with highest weight $(r-1,1)$ is a summand of the $r$th Lie power if and only if $r$ is a not power of $p$.